modelling of physical processes in ion lithography

144 Thin Solid Films, 214 (1992) 144 149

Modelling of physical processes in ion lithography

K. V u t o v a a n d G . M l a d e n o v Institute of Electronics, Bulgarian Academy of Sciences, blvd. Trakia 72, Sofia 1784 (Bulgaria)

(Received November 5, 1991; accepted January 24, 1992)

Abstract

In this paper a complete mathematical model for exposure and development process simulations in ion lithography is described. The corresponding software package is developed. The main steps of the model are (1) exposure process modelling, involving ion scattering simulation and dissipated energy calculation, and (2) development process modelling, involving solubility rate calculation and the simulation of developed profile evolution. The presented models are realized by means of a software package. Thus, it is possible to predict the final result in the case of ion lithography under given initial conditions. The simulation result is very useful when optimizing particular technological processes in microcircuit lithography. It can be used for the development of an algorithm to correct the proximity effect of real microelectronic structures.

1. Introduction

The process of lithography involves the formation of patterns for selective area processing of devices at different stages of their fabrication. While conventional lithography is carried out using light for exposing the "resist", the continuing miniaturization of integrated circuits has stimulated interest in new exposure tech- niques. Electrons, X-rays and ion beams can also de- posit energy in a resist to expose it. Ion beams offer ultimate advantages in sensitivity and fineness of feature size because of their particular penetration proper- ties. The resolution in this case is intrinsically higher than in the case of electron beams because the electrons suffer from the "proximity" effect [1 3]. The lateral scattering of electrons and the creation of energetic secondary electrons both complicate and ultimately limit the patterns that have very fine dimensions. Pri- mary electron scattering and secondary electron pro- duction from one feature tend to spear in a neigh- bouring feature with a loss of contrast between them. Ions scatter much less and produce secondary electrons of only very low energy, thus reducing the spreading of exposure features in a resist to less than 100 A [4-6].

The great capabilities of ion lithography with respect to practical applications necessitate detailed studies of their behaviour and the development of comprehensive mathematical models of the physical processes in- volved. There exist two kinds of models to simulate the penetration of energetic particles into targets: analytical models and numerical models. The analytical models are based on some model approximations and their application is bounded by a set of strong limitations and constraints (one-component target, point source

etc.), i.e. the analytical method's results are not always applicable for real experiments. The numerical methods are usually based on the Monte Carlo technique. To achieve a reasonable certainty of statistical results, one has to simulate a great number of trajectories, which is a long-time and memory-consuming process.

In this paper we describe a complete mathematical model and the corresponding software for exposure and development process simulations related to ion lithography. It allows the accuracy of the result to be im- proved considerably without increasing the number of simulated trajectories. The main steps of the exposure modelling process are ion scattering simulation and dissipated energy calculation. The development process modelling includes solubility rate calculation and simulation of developed profile evolution.

2. Exposure process modelling

The modelling of the exposure process in ion lithography involves the simulation of ion penetration into materials as well as the calculation of the absorbed energy distribution within the targets. In the following subsections corresponding mathematical models are proposed and their basic features are described. Appro- priate results of the modelling are also presented and discussed.

2.1. Ion scattering in amorphous targets The main computer program for ion scattering in

amorphous materials using Monte Carlo calculations is TRIM, first published by Biersack and Haggmark [1], but later extended fully to cover collision cascades

0040-6090/92/$5.00 :(! 1992 - - Elsevier Sequoia. All rights reserved

K. Vutova, G. Mladenov / Modelling of ion lithography 145

including the motion of recoils [2]. The program is by far the most rapid at present. For the particular energy range in ion lithography, the calculations are further speeded up by introducing an approximate but rather accurate treatment of the individual collision process. The program has spread in a large number of versions over the entire field of atomic collisions in solids. It was applied for the first time to the field of ion lithography in ref. 3.

We have developed a program version to model the ion scattering in multilevel multicomponent amorphous targets, named TRIM-MVD [7]. The penetration of each particle from one statistically significant extract (about 1000 particles) is considered as an ele- mentary collision sequence with the target atoms. We calculate the characteristic changes in the particle motion for each collision, assuming a straight line trajec- tory between two collisions. The program involves a Monte Carlo technique to calculate the kind of the atom taking part in the collision, the azimuthal angle value, the impact parameter value etc. A detailed de- scription and the program results are presented else- where [7, 8].

2.2. Exposure intensity calculation in the case of ion bombardment

To develop the corresponding exposure correction methods and the development profile evolution simulation models, it is necessary to know the dissipated energy space distribution in the resist, when exposing arbitrary points and real microimages.

The next subsections deal with the model developed for improvement of the accuracy of the absorbed energy distribution at large lateral distances during the exposure of a single point of the resist surface. A three-dimensional model for calculation of the absorbed energy distribution in the case of an arbitrary pattern exposure is presented.

2.2.1. Accuracy improvement of the absorbed energy distribution at large radial distances To achieve a satisfactory statistical accuracy, using

Monte Carlo calculations, it is often necessary to simulate a large number of trajectories. The particle trajec- tory density near the beam axis is greater than in distant regions. This is why satisfactory calculation accuracy near the beam axis is achieved by tracing a few particle trajectories ( ( 2 - 3 ) x 103). It should be pointed out that the increase in the number of trajectories being modelled, with the purpose of producing statistical consistency for large lateral distances (characterizing the backward-scattered particles and giving the greatest contribution to the proximity effect), is not quite effective, as only few trajectories travel through these regions.

We have developed a Monte Carlo methodology and a corresponding computer program called BET-MK to solve the problem concerning the insufficient statistics of the discrete data for the absorbed energy in the case of large lateral distances. The main features of this methodology are as follows.

(1) The numerical data for the discrete space distribution of the absorbed energy are transformed into analytical functions. The data arrays containing the energy distribution when one point from the resist surface is exposed are obtained using the TRIM-MVD computer program.

(2) The absorbed energy at some resist depth is approximated as a sum of two gaussians:

f ( r ) = k exp - -~)+ t /E~bbeXp~ (1)

where k is a normalization constant, /?f and /~b a re the characteristic widths of the forward and the backward scattering particles and r/E is the ratio of the energy depth dissipation of the backward scattering particles to that of the forward scattering particles. The input data for the program BET-MK are the two-dimensional arrays containing the absorbed energy distribution values obtained as a result of the modelling of the trajectories.

(3) The parameter values (/?f, flU, qE) are calculated using an original Monte Carlo technique, instead of the non-linear least-squares method and an arbitrary kind of distribution. The technique comprises mean square deviation minimization by the decrement of interval lengths for each of the parameters chosen. The minimization is carried out in an iteration loop.

2.2.2. Three-dimensional model for the absorbed energy calculation To obtain the absorbed energy space distribution

when exposing an arbitrary pattern, using an arbitrary exposure dose distribution, one must take into account the influence of a large number of exposed points. The absorbed energy space distribution in the case of an arbitrary pattern can be calculated by integrating the data obtained by the computer simulation. Because of the large number of calculations, a simplified procedure should be used to calculate the integral space distribution of the absorbed energy. The main features of the procedure proposed in ref. 9 are as follows.

(1) The two-dimensional data array containing the absorbed energy values at some resist depth is presented as eqn. (1). If the ion exposure is uniformly distributed over an area A, then the energy density can be expressed as

F(r) = i f ( r ) dA (2) Id

A

146 K. Vutova, G. Mladenov / Modelling q f ion lithography

I f the area A is a simple pattern (i.e. a line or a rectangle) the integral (2) can be calculated using the tabulated error function:

l

i(::) eft(t, a) = exp - dx (3)

0

(2) In the case of a more complex pattern, the area should be divided into simple parts and then the corresponding values of the absorbed energy should be subsequently summed. The obtained formulae for the absorbed energy density, when exposing a line, a line segment or a rectangle, are given in ref. 9. These simple patterns are sufficient to compose an arbitrary figure.

(3) The procedure takes into account the radial variation in the absorbed energy as well as its modification vs. the depth of the resist. To calculate the values of fir, fib and r/e, a linear approximation along the resist depth is used. In some ion lithography cases (heavy ion bombardment , high energy ion beams etc.), it is necessary to use more than two layers along the resist depth. The values of fir, fib and v/v_ change linearly among them.

2.2.3. Results and discussion The main advantages of the proposed model are the

following. (1) The parameters fir, fib and q~. in formula (1) have

direct physical meaning. The formula 's presentation as a sum of two gaussians makes it simple and easy to use in further mathematical transformations.

(2) The developed Monte Carlo method does not allow the possibility of an infinite loop in the case of a local minimum that is the case in some of the least- squares methods.

(3) The use of analytical functions instead of numerical discrete absorbed energy data has the following benefits: (a) significant improvement of the statistical accuracy of the absorbed energy in the case of large lateral distances and one-point exposure; (b) formula (2) gives a more convenient and accurate way to calculate the integral energy space distribution in the case of an arbitrary pattern exposure, instead of the complex, time-consuming and error-propagating matrix opera- tions when using discrete data; (c) the analytical expres- sion allows the energy modification vs. the resist depth to be taken into account in the exposure and development process modelling.

Typical results obtained by the exposure simulation are shown in Fig. 1. The parameters fir, [Jb and ~/E are determined using the absorbed energy data arrays obtained by the computer program T R I M - M V D . Figure l shows the lateral distribution of the absorbed energy at three depths: 1040, 2000 and 4000 A for an incident 6 line of 60 keV hydrogen ions in 4000/~ thick poly- methylmethacrylate (PMMA) on silicon. The spread of

! 4000 400

~ 0 0 ~' 4oaa

'°

4 ~ ,~o 40 6'0 ~o " g'O 60 ~--

eo teFo~ d/5tonc& nm

Fig. I. Lateral variation in absorbed energy at three characteristic depths ( 1040 ~,, 2000/~ and 4000/~) for a line of hydrogen ions.

(a) a¢oowe numSer (b) otormc number

Fig. 2. Dependence of (a) fi~ and (b) fib on the resist depth when exposing with 200 keV beryllium ions, 200 keV silicon ions and 50 keV gallium ions.

the absorbed energy along the depth is small (practi- cally negligible).

The values of the parameters fir and ~b as a function of the resist depth when exposing bulk P M M A to different ions are shown in Fig. 2. In the triangular diagram the points corresponding to the values of []r or fib, the resist depth, and the atomic number represent vertices of inscribed triangles. Using these results one can approximately determine the fir and [~b values for different resist depths and atomic numbers (when the beam energy is fixed). For instance, for 200 keV beryllium ions incident on bulk PMMA, the fib value at a depth of 0.7 lam is in the range 0,033 0.08 lam. For 200 keV ions with atomic number between 4 and 14, the value of the parameter [2~ b a t a depth of 0.5 pm is in the range 0.034-0.044 pm. Similarly, if fir (or fib) values and the atomic number are known, one can determine the resist depth and, vice versa, from known fir (or fib) values and the resist depth, the atomic number can be determined (when the beam energy is fixed).

K. Vutova, G. Mladenov / Modelling o1" ion lithography 147

3. Development process modelling

In the case of high resolution ion lithography it is very important to know the developed line profiles instead of the latent image in the resist; as well as the basic dependences, characterizing the development process in terms of the developed resist depth as a function of the exposure dose and the developing time, developing time as a function of the exposure dose etc. The profile shape often has a direct influence on the subse- quent processes in optical and electron device manufac- ture.

The resist development characteristics are expressed by the solubility rate. This is the most important parameter in the development process study. We have described in ref. 10 a methodology for calculating the unknown parameter values which determine the solubility rate for an arbitrary couple of positive resist-developer in electron and ion lithography.

A mathematical model is described in ref. 11 for two-dimensional and three-dimensional computer simulations of the dynamics of polymer topological struc- ture development in electron and ion lithography. The physical resist surface and the processes occurring between the developer and the resist are approximated using the pattern macromodifications and the mathematical motion of the polymer surface. This is based on the investigation of the developed contours evolution in a chosen grid of points. We assume that the resist is homogeneous enough and that the development process is isotropic. The main features of the proposed model are as follows.

(1) The motion of the evolving points takes place along the normals to the corresponding profiles.

(2) The space modification of the absorbed energy distribution is also taken into account.

(3) In our model we use a cubic spline in the two-dimensional case and a bicubic spline in the three-dimensional case to describe the developed profiles.

(4) An original procedure is applied for the profile discretization to increase the contour accuracy. The procedure goal is achieved by decreasing the number of evolving points in the parts of the profile whose normal tilt value is very close to that of the initial profile, and increasing the number of evolving points in the parts that have a significantly modified tilt.

(5) The contour accuracy obtained for a given time can be increased by decreasing the time step if necessary.

3,1. Results and discussion The main advantage of the proposed model is that it

can be easily adapted for three-dimensional computer simulation in comparison with the cell removal [ 12] and string [12] models.

The use of the spline functions allows precise calcula-

L,

I ) Jl..8 .,u.rl ~ J e.,,~ .u.r~

Fig. 3. Two-dimensional pattern used for illustration purposes. All small gaps are 0.1 gm.

Fig. 4. Computer simulation results for a developed surface using 1:3 solution of methylisobutylketone (M1BK) and isopropyl alcohol (IPA), applying the ideal pattern shown in Fig. 3.

tion of the normal in the corresponding evolving point to be made.

A general test pattern is shown in Fig. 3. The pattern was chosen for the difficulties it poses to the litho- graphic process (the interaction of the topological structures at short distance and the influence of backward scattering particles). Developed profiles obtained by computer simulation using this test pattern are shown in Fig. 4 and Fig. 5. The beam (60 keV hydrogen ions) is gaussian distributed (eqn. (1)). The three- dimensional profile shown in Fig. 4 represents the developed gap surface at a dose of 2 x 10 VCcm 2. The development time is 0.25 min. The developed contours at various resist depths are shown in Fig. 5. This shows the abrupt change in the developed gap walls. Hence, one can obtain a higher contrast by changing the development time and the exposure dose. Sufficient contrast at higher doses can be obtained for shorter times.

Figure 6 shows the modification of the mean developed line width as a function of the exposure parameter Q/Qo (Q is the exposure dose, Q0 is the initial exposure

148 K. Vutova, G. Mladenov / Modelling of ion lithograph)'

i| i | | ! ! ! ! I | | | I | | I I l I I I I

i

I

l U

1 0 4 ~

m

I m

10400.00

IOMO, O0 111~0.00 r~OOJO0 ~QO0,O0 4~lOOJO0 _a_~y~_~_ 01_m~__,~_

Fig. 5. Developed contours at various resist depths: 200/~, 1200/~, 2200/~ and 3200 A.

0./ # 0.o5

0.o¢

= O. f125/.~rn

.. 0. 0625/um

~= 0.0225/.zm

f o to 2o j o - 4"o ~o a / d o

Fig. 6. Variation in the main developed line width vs. the exposure parameter Q/Qo, Qo = I x 10 7 C cm 2. The development time is 15 s. The resist thickness is 0.4 I'm on silicon, with exposure to a 60 keV hydrogen ion beam. , results using as the developer a 1:3 solution of MIBK and IPA; - , using MIBK.

oo;

o o t

¢

~, 0 0625t.tr~

0 to - 20 50 --- ~-~o 5"0 0/~o

Fig. 7. Variation in the developed line width at three characteristic depths: curve 1, the developed line width at a depth of ZOO ~; curve 2, at a depth of 2000 ~: curve 3, at a depth of 4000 ~. MIBK:IPA (1:3) developer.

dose) when using different widths of the exposed lines (from 0.0225 pm to 0.1125 pm). The mean developed line widths are calculated as a mean value of the developed profile widths at three characteristic depths: 200/~, 2000 A and 4000 A. These results

are obtained when exposing 0.4 gm of PMMA on silicon to hydrogen ions at 60 keV. The initial dose Q 0 = l x 10 7 C c m -2. In Fig. 7, the curves represent the developed line width dependences at depths of 200, 2000 and 4000/~ and l = 0.0625 pm. One can conclude

K. Vutova, G. Mladenov / Modelling of ion lithography 149

that the less powerful developer MIBK:IPA (1:3) is more convenient: the developed line width is closer to the projected line width. In this case we observe a stabiliza- tion of the developed line width modificatons when the exposure parameter value is between 6 and 8. The developed profiles have abrupt walls (Fig. 7), i.e. the contrast is the best, preserving the projected line width during the development process for these doses. This result enables the optimal exposure and development conditions to be chosen for the desired profile formation. One can see that (i) the developed width modification is monotonic and (ii) the broadening is not significant when using the powerful developer MIBK. This is due to the small coefficient of backward substrate reflection which is a crucial factor for the proximity effect.

4. Conclusions

during the modelling of the exposure and development processes.

A mathematical model is developed for two-dimensional and three-dimensional computer simulations of the development dynamics in a polymer. The presented models for the exposure and development process simulation are realized by means of a software package. Thus it is possible to predict the final result in the case of ion lithography under given initial conditions. The simulation result is very useful when optimizing particular technological processes in microcircuit lithography. The model described and the software package developed for the exposure and development process simulations in ion lithography can be used to elaborate an algorithm for the correction of the developed topology as a result of the proximity effect in real microelectronic structures.

In this paper we have described the basic steps when modelling the ion scattering process in multilevel multicomponent amorphous targets. The problem concerning the insufficient statistics of the discrete data for the absorbed energy in the case of large lateral distances is overcome. A Monte Carlo methodology is developed for the transformation of the numerical data array, representing the absorbed energy space distribution when exposing one point from the resist surface, into the form of analytical functions. A methodology is proposed for the calculation of the absorbed energy integral space distribution, when exposing real topological structures in ion lithography, using analytical ex- pressions. The use of analytical functions improves significantly the statistical accuracy of the absorbed energy data in the case of large lateral distances, gives more time and a memory-cheaper way to calculate the integral energy space distribution when using an arbitrary pattern exposure, and allows the energy modification along the resist depth to be taken into account

References

1 J. P. Biersack and L. G. Haggmark, Nucl. lnstrum. Methods, 174 (1980) 257,

2 J. P. Biersack and W. Eckstein, Appl. Phys. A, 34 (1984) 73. 3 G. Mladenov, M. Braun, B. Emmoth and J. P. Biersack, J. Appl.

Phys., 58 (7) (1985) 2534. 4 R. L. Kubena, J. Y. Lee, R. L. Jullens, R. G. Brault, P. L.

Middlenot and E. H. Stevens, IEEE Trans. Electron Deviees, 31 (1984) 1186.

5 T. Shiokawa, Y. Aoyagi, P. H. Kim, K. Toyoda and S. Namba, Jpn. J. Appl. Phys., 23 (1984) 232.

6 I. Adesida and L. Karapiperis, J. Appl. Phys., 56 (1984) 1801. 7 G. Mladenov, Proc. Int. Conf. on Electron Beam Technologies,

Varna, Bulgarian Academy of Sciences, Sofia, 1985, p. 23. 8 G. Mladenov, DSc Thesis, Institute of Electronics, Bulgarian

Academy of Sciences, Sofia, 1986. 9 K. Vutova and G. Mladenov, J. Inf. Rec. Mater., Germany, 19 (4)

(1991) 273. 10 K. Vutova and G. Mladenov, Thin Solid Films, 200 (1991) 353. 11 K. Vutova and G. Mladenov, J. Inf. Rec. Mater., /9(4) (1991) 283. 12 R. E. Jewett, P. I. Hagouel, A. R. Neureuther and T. Van Duzer,

Polym. Eng. Sci., 17 (6) (1977) 381.

modelling of physical processes in ion lithography

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